What Is the Complexity of Surface Integration?

We study the worst case complexity of computing ε-approximations of surface integrals. This problem has two sources of partial information: the integrand f and the function g defining the surface. The problem is nonlinear in its dependence on g. Here, f is an r times continuously differentiable scalar function of l variables, and g is an s times continuously differentiable injective function of d variables with l components. We must have d ≤ l and s ≥ 1 for surface integration to be well-defined. Surface integration is related to the classical integration problem for functions of d variables that are min{r, s − 1} times continuously differentiable. This might suggest that the complexity of surface integration should be2((1/ε)d/min{r,s−1}). Indeed, this holds when d < l and s = 1, in which case the surface integration problem has infinite complexity. However, if d ≤ l and s ≥ 2, we prove that This research was supported in part by the National Science Foundation under Grant CCR9987858. †This research was supported in part by the National Science Foundation under Grant CCR9731858.